Affine Isoperimetric Inequalities And Minkowski Type Problems In Brunn-Minkowski Theory

The researches of this thesis belong to convex geometric analysis and are de-voted to the study of affine isoperimetric inequalities and Minkowski type prob-lems in the Brunn-Minkowski theory.Affine isoperimetric inequalities mainly focus on the extremal problems of some affine invariants of convex bodies and are widely used in information theory and analytical inequalities.The Minkowski problem is commonly concerned by convex geometric analysis,partial differential equations,differential geometry and other disciplines,which mainly concentrates on the existence,uniqueness and regularity of its solution.The solution of the Minkowski problem has become a powerful tool in establishing some analytic in-equalities.The research contents associated with affine isoperimetric inequalities and Minkowski type problems are divided into the following six partsPart ?.In 2014,Xi,Jin and Leng defined the Orlicz addition of convex bodies in the Orlicz Brunn-Minkowski theory,and established the Orlicz Brunn-Minkowski inequality by using Steiner symmetrization.In chapter 2,we give a new proof for the Orlicz Brunn-Minkowski inequality based on the shadow system.Part ?.In 2018,Lutwak,Yang and Zhang defined(p,q)-mixed volume.We introduce the concept of(p,q)-mixed geominimal surface area based on the(p,q)-mixed volume,and establish the related affine isoperimetric inequalities.Part ?.We propose the Orlicz Aleksandrov problem in the Orlicz Brunn-Minkowski theory.In the case of even measure,the solutions of two different conditions of the problem are given by variational method.Part ?.Zou and Xiong proposed the Lp q-capacitary Minkowski problem,and completely solved the problem when p>1 and 1<q<n.Xiong et al.further obtained the solution to the problem for discrete measure when 0<p<1 and 1<q<2.In chapter 5,when 0<p<1 and 1<q<n we prove the existence of solutions to this problem for general measure.Part V.The logarithmic q-capacitary Minkowski problem is very important in the Lp q-capacitary Minkowski problem.In chapter 6,we give the solution to the problem for discrete measure.Part VI.Hong,Ye and Zhang studied the Orlicz q-capacitary Minkowski problem and obtained the solution of this problem involving p>1 and 1<q<n.In chapter 7,we give the solution of the problem involving p<0 and 1<q<n for discrete measure.